Mathematics for the interested outsider

Given two finite-dimensional vector spaces and , with bases and respectively, we know how to build a tensor product: use the basis .

But an important thing about the tensor product is that it’s a functor. That is, if we have linear transformations and , then we get a linear transformation . So what does this operation look like in terms of matrices?

First we have to remember exactly how we get the tensor product . Clearly we can consider the function . Then we can compose with the bilinear function to get a bilinear function from to . By the universal property, this must factor uniquely through a linear function . It is this map we call .

We have to pick bases of and of . This gives us a matrix coefficients for and for . To calculate the matrix for we have to evaluate it on the basis elements of . By definition we find:

that is, the matrix coefficient between the index pair and the index pair is .

It’s not often taught anymore, but there is a name for this operation: the Kronecker product. If we write the matrices (as opposed to just their coefficients) and , then we write the Kronecker product .

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.